How to Calculate Orders of Magnitude in Exponents

Converting a number to an order of magnitude displays its approximate scale and omits any absolute measurement, which means it has zero significant figures. The resulting number readily compares to other figures to estimate the magnitude of their differences. Order of magnitude uses a base-10 logarithmic scale, so each step on the scale is 10 times the previous one, or 10 to the power of the magnitude. As an example, saying a man is three orders of magnitude taller than an ant means he's approximately 1,000 times taller (1, followed by three "magnitude" zeros).

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Instructions

Convert figures to a common unit. It doesn't matter what that unit is, but each figure should use the same one. As an example, you might say a man is 1.7 meters tall, but a 2-mm ant is 0.002 meters tall.

Express each figure in scientific notation by writing it with one digit before the decimal point, followed by "x 10^n" where "n" is the number of places the decimal moved. Moving the decimal to the left results in a positive number, but moving it to the right results in a negative number. In the example, you didn't have to move the decimal for 1.7, so the n is zero, giving you "1.7 x 10^0." However, you had to move the decimal in the ant's height three places to the right, so n is negative three, giving you "2.0 x 10^-3."

Round the figure to the closest power of 10. Because we're rounding scientific notation, you must use the square root of 10 (3.16) as the rounding cut-off, instead of the arithmetic five. Using this cut-off results in a 3-factor maximum scale error, instead of the arithmetic method's potential error factor of 10. In the example, both numbers are less than 3.16, so the orders of magnitude are simply 10^0 and 10^-3, respectively. However, if you had the number 4.1 x 10^5, 4.1 is larger than 3.16, so round up to 10 and then move the decimal point again; therefore, the 10^5 part increments one to describe a 10^6 order of magnitude.

Subtract the smaller magnitude exponent from the larger one to calculate the scale difference. In the example, 0 minus negative 3 calculates a difference with three orders of magnitude.

Tips & Warnings

If you have a scientific calculator, enter the full number, press "log" and round to the nearest integer to get the order of magnitude for the number. Because the "log" function already considers the 10-root, use 5 as the rounding cut-off. As an example, if you entered 410,000 and pressed "log," you'd get 5.61, so round up to 6 orders of magnitude. Some researches simply truncate this number to five orders of magnitude to describe the number between five and six orders of magnitude, but rounding produces a more accurate estimate.

To compare figures using a scientific calculator, divide the larger value by the smaller value, press "log" and then round the result. In the man-ant comparison, divide 1.7 by 0.002 to get 850, and then press "log" to get 2.93. Rounding up describes a difference with three orders of magnitude. However, you don't always need to round the number; many frequently used scales, such as the Richter scale and scales for decimals of sound and pH acidity, use real numbers to describe comparative magnitudes.